# What is Riemann tensor: Definition and 74 Discussions

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. It assigns a tensor to each point of a Riemannian manifold (i.e., it is a tensor field). It is a local invariant of Riemannian metrics which measure the failure of second covariant derivatives to commute. A Riemannian manifold has zero curvature if and only if it is flat, i.e. locally isometric to the Euclidean space. The curvature tensor can also be defined for any pseudo-Riemannian manifold, or indeed any manifold equipped with an affine connection.
It is a central mathematical tool in the theory of general relativity, the modern theory of gravity, and the curvature of spacetime is in principle observable via the geodesic deviation equation. The curvature tensor represents the tidal force experienced by a rigid body moving along a geodesic in a sense made precise by the Jacobi equation.

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1. ### Problems with the Riemann tensor in general relativity

After Taylor expansion and using equations (2), I have no problem getting to equation (1). Now obviously I have to somehow use (3.71) ,which I do know how, to derive to express the second order derivative. On the internet I found equation (3), and I have tried to understand where this comes from...
2. ### I Ricci notations and visualisation

I'm having trouble with notations and visualisations regarding Ricci curvature. For Riemann tensor there is variously: ##R^{\rho}\text{ }_{\sigma\mu\nu}\text{ }X^{\mu}Y^{\nu}V^{\sigma}\partial_{\rho}## ##[\nabla _{X},\nabla _{Y}]V## ##R(XY)V\mapsto Z## ##\left\langle R(XY)V,Z...
3. ### I First algebraic Bianchi identity of Riemann tensor (cyclic relation)

I am guessing that: $R_{a[bcd]}=0$ can not be derived from the symmetries of $R_{ab(cd)]}=R_{(ab)cd}=0$ $R_{[ab][cd]}=0$ ?Sorry when I search the proof for it I can not find much, it tends to come up with the covariant Bianchi instead. I am guessing it will need one of the symmetries above...
4. ### A GW Binary Merger: Riemann Tensor in Source & TT-Gauge

In the book general relativity by Hobson the gravitational wave of a binary merger is computed in the frame of the binary merger as well as the TT-gauge. I considered what components of the Riemann tensor along the x-axis in both gauges. The equation for the metric in the source and TT-gauge are...
5. ### I Calculate Contraction 2nd & 4th Indices Riemann Tensor

How to calculate the contraction of second and fourth indices of Riemann tensor?I can only deal with other indices.Thank you！
6. ### Does anyone know which are Ricci and Riemann Tensors of FRW metric?

I just need to compare my results of the Ricci and Riemann Tensors of FRW metric, but only considering the spatial coordinates.
7. ### I A couple questions about the Riemann Tensor, definition and convention

According to Wikipedia, the definition of the Riemann Tensor can be taken as ##R^{\rho}_{\sigma \mu \nu} = dx^{\rho}[\nabla_{\mu},\nabla_{\nu}]\partial_{\sigma}##. Note that I dropped the Lie Bracket term and used the commutator since I'm looking at calculating this w.r.t. the basis. I...
8. ### I Calculating Covariant Derivative of Riemann Tensor in Riemann Normal Coordinates

Hello everyone, in equation 3.86 of this online version of Carroll´s lecture notes on general relativity (https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll3.html) the covariant derviative of the Riemann tensor is simply given by the partial derivative, the terms carrying the...
9. ### I Riemann Tensor, Stoke's Theorem & Winding Number

I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve. If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?
10. ### A Riemann Tensor Formula in Terms of Metric & Derivatives

Could someone please write out or post a link to the Riemann Tensor written out solely in terms of the metric and its first and second derivatives--i.e. with the Christoffel symbol gammas and their first derivatives not explicitly appearing in the formula. Thanks.
11. ### I Computing Riemann Tensor: 18 Predicted Non-Trivial Terms

I want to compute the Riemann Tensor of the following metric $$ds^2 = dr^2+(r^2+b^2)d \theta^2 +(r^2+b^2)\sin^2 \theta d \phi^2 -dt^2$$ Before going through it I'd like to try to predict how many non-trivial components we'd expect to get, based on the Riemann tensor basic rule: It is...
12. ### I Riemann Tensor: Questions & Geometric Interpretation

Tensor of Riemann. Geometric interpretation.The Riemann tensor gives the variation of a vector displaced parallel in a closed loop, say a small rectangle formed by geodesic sides, (δa) and δb) first, starting from a vertex A and going to another vertex in the diagonal, B; then starting from the...

41. ### Using parallel propagator to derive Riemann tensor in Sean Carroll's

Hello all, In Carroll's there is a brief mention of how to get an idea about the curvature tensor using two infinitesimal vectors. Exercise 7 in Chapter 3 asks to compute the components of Riemann tensor by using the series expression for the parallel propagator. Can anyone please provide a...
42. ### Why Do These Riemann Tensor Terms Cancel Each Other Out?

I was working on the derivation of the riemann tensor and got this (1) ##\Gamma^{\lambda}_{\ \alpha\mu} \partial_\beta A_\lambda## and this (2) ##\Gamma^{\lambda}_{\ \beta\mu} \partial_\alpha A_\lambda## How do I see that they cancel (1 - 2)? ##\Gamma^{\lambda}_{\ \alpha\mu}...
43. ### What is the Form of Riemann Tensor in 3D?

hi Riemann tensor has a definition that independent of coordinate and dimension of manifold where you work with it. see for example Geometry,Topology and physics By Nakahara Ch.7 In that book you can see a relation for Riemann tensor and that is usual relation according to Christoffel...
44. ### Calculate the elements of the Riemann tensor

Homework Statement Compute 21 elements of the Riemann curvature tensor in for dimensions. (All other elements should be able to produce through symmetries) Homework Equations R_{abcd}=R_{cdab} R_{abcd}=-R_{abdc} R_{abcd}=-R_{bacd} The Attempt at a Solution I don't see how 21...
45. ### Local flat space and the Riemann tensor

Can somebody explain in layman's terms the connection between local flat space (tangent planes) on a manifold and the Riemann tensor.
46. ### Riemann tensor cyclic identity (first Bianchi) and noncoordinate basis

I got trouble to understand the cyclic sum identity (the first Bianchi identity) of the Riemann curvature tensor: {R^\alpha}_{[ \beta \gamma \delta ]}=0 or equivalently, {R^\alpha}_{\beta \gamma \delta}+{R^\alpha}_{\gamma \delta \beta}+{R^\alpha}_{\delta \beta \gamma}=0. I can understand the...
47. ### Calculation of double dual of Riemann tensor

Hi all, I encounter a technical problem about tensor calculation when studying general relativity. I think it should be proper to post it here. Riemann curvature tensor has Bianchi identity: R^\alpha_{[\beta\gamma\delta;\epsilon]}=0 Now given double (Hodge)dual of Riemann tensor: G = *R*, in...
48. ### Interpretation of torsion vs riemann tensor

Hi all, I am working through Visser's notes http://msor.victoria.ac.nz/twiki/pub/Courses/MATH465_2012T1/WebHome/notes-464-2011.pdf section 3.5 onward. I am trying to differentiate between the torsion and the Riemann curvature tensor in a heuristic manner. It appears from "Geometric...
49. ### Calculating the Riemann Tensor

The work I have been following has me very confused... and I am almost sure I am making a mistake somewhere! After working up to this equation: \delta V = dX^{\mu}\delta X^{\nu} [\nabla_{\mu} \nabla_{\nu}]V I am asked to calculate the curvature tensor. Now the way I did it, turned out...
50. ### No. of Independent Components of Riemann Tensor in Schwartzchild Metric

In general 4d space time, the Riemann tensor has 20 independent components. However, in a more symmetric metric, does the number of independent components reduce? Specifically, for the Schwartzchild metric, how many IC does the corresponding Riemann tensor have? (I think it is 4, but I...